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REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
The numbers in brackets are assigned according to the American Mathematical
Society classification scheme. The 1980 Mathematics Subject Classification (1985
Revision) can be found in the December index volumes of Mathematical Reviews.
20(65-01, 65Mxx, 65Nxx].—MYRON B. ALLEN III, ISMAEL HERRERA &
GEORGE F. Pinder, Numerical Modeling in Science and Engineering, Wiley,
New York, 1988, x+418 pp., 24 cm. Price $39.95.
This book seeks to give a unified treatment of numerical modeling by combining
material from continuum mechanics, partial differential equations (PDE's) and nu-
merical methods for PDE's. The book is intended to serve as a text for a year-long
graduate course for engineers, scientists and applied mathematicians. The basic
idea of the book, namely, to first derive the equations from physical principles,
then consider fundamental mathematical questions such as existence and qualita-
tive properties of solutions, and finally devise numerical methods (finite difference
and finite element methods), is of course sound. It should help the student to gain
a better understanding of numerical methods if a proper theoretical background
is given, and the theoretical studies are well motivated if the usefulness and im-
portance is shown of numerical modeling in applications. However, such a unified
approach requires a very careful selection of material to be successful.
The present book starts with a chapter where the basic PDE's in continuum
mechanics (fluid mechanics and elasticity) are derived. Then there follow an in-
troductory chapter on numerical methods and three chapters on steady state (el-
liptic), dissipative (parabolic) and nondissipative (hyperbolic) problems involving
both mathematical and numerical aspects. In the final chapter some nonlinear
problems are considered. The style of the book is informal or "nonmathemati-
cal"; for example, no theorems with precise assumptions are stated. To my taste,
the mathematical level of the text is too low to be really suitable for a graduate
course. For instance, variational principles for Poisson's equation and finite ele-
ments are discussed without the notions of the H1-norrci and completeness, which
seems unnecessarily restrictive, since these concepts are now common also in the
engineering literature. Mostly standard material concerning finite difference meth-
ods is presented in the chapters on elliptic, parabolic and hyperbolic problems. The
book also contains material on finite element methods for these problems, but here
the presentation is not adequate. For instance, in Chapter 2, second-order differen-
tial operators are applied to piecewise linear functions which are only continuous,
and it is stated that the error in the energy norm using piecewise linear basis func-
tions is 0(h2). Such things should be very confusing to the student and defeat
the stated ambition of the authors to increase the understanding of the subject.
©1989 American Mathematical Society0025-5718/89 $1.00 + $.25 per page
761
762 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
Chapter 5 contains a section on finite elements for hyperbolic problems, which is a
very active area of research, where rapid progress has been made in the last years,
for example on problems in several dimensions and nonlinear problems. However,
this development is not touched upon in the present book.
To sum up, I find the basic idea of the book to be very natural but I think
it should be possible to give a presentation which would be more precise math-
ematically and more up-to-date numerically, without becoming more difficult to
read.
Claes Johnson
Department of Mathematics
Chalmers University of Technology
and University of Göteborg
S-412 96 Göteborg, Sweden
21[65-02, 76-02].—OLIVIER PiRONNEAU, Méthodes des Éléments Finis pour les
Fluides, Collection Recherches en Mathématiques Appliquées, Vol. 7, Masson,
Paris, 1988, 199 pp., 24 cm. Price FF 175.00.
This short monograph appears in a relatively new collection edited by P. G.
Ciarlet and J.-L. Lions. Although the collection accepts texts in both French and
English, most titles are in French. Such an outlet for high-quality research is
especially welcome for Ph.D. level teaching where there is a definite lack of this
kind of material. In this respect, the book by O. Pironneau is exemplary. It
presents an excellent summary of classical problems of fluid mechanics, including
a discussion of the physical limitations of models. It also presents a fairly large
number of methods for convection-diffusion problems which are probably the most
difficult ones in fluid simulation at high Reynolds number. The results are in no
way complete, but they leave the reader with a real sense of what the issues are.
The treatment of incompressibility is well done, even if the repertoire of elements
is a little short. There is a treatment of less standard boundary conditions, which
can also be very useful for beginners; this kind of material is alway very hard to
find.
Navier-Stokes equations, both for incompressible and compressible flow, are con-
sidered. Again one should not look for complete results but for a broad picture
sketching the main facts.
The book contains a collection of information which can nowhere else be found
in one place. It should definitely be compulsory reading for beginners in flow simu-
lation. Experienced readers will enjoy the practical presentation and will discover
links between facts and methods they may not have thought about before.
Michel Fortin
Département de Mathématiques
Université de Laval
Cité Universitaire
Quebec, Quebec G1K 7P4, Canada
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 763
22(65-04, 65M60, 65N30]. I. M. SMITH & D. V. GRIFFITHS, Programming
the Finite Element Method, 2nd ed., Wiley, Chichester, 1988, xii+469 pp., 23^
cm. Price $49.95.
The major objective of this text is to present the reader with an understanding of
how finite element concepts are realized in computer software. This is accomplished
rather effectively by means of a software "building block" technique whereby rel-
atively complex programs may be assembled from a library of relatively simple,
special purpose subroutines, each capable of carrying out some important step in
the finite element analysis process. This device allows the reader a rather clear
view of the overall process, since programs can be written quite compactly making
use of building block subroutines.
The author very thoughtfully starts out by assembling subroutines into relatively
narrow, simple programs and progresses to broader, more complex programs by
modifications and additions.
The building block subroutines are categorized as either "black box" routines or
"special purpose" routines. Black box routines are those associated with standard
matrix handling procedures, and, as the name implies, little explanation of the
bases of these routines is provided. Special purpose routines, on the other hand,
accomplish some significant step in finite element analysis. Although a theoreti-
cal basis of each of these routines is presented, these discussions often seemed too
abbreviated to provide adequate background for the finite element novice. It is
recommended that reading of this text either be preceded by reading of an intro-
ductory finite element text or accompanied by a thorough classroom presentation
of the finite element method.
A number of example programs are presented in the various chapters of the text.
These presentations are enhanced by the availability of a summary of the functions
of each special purpose subroutine in an appendix and a listing of each of these
subroutines in another appendix. The reader may have to bounce back and forth
in these various areas of the text as he/she reads, but this becomes relatively easy
as familiarity with the arrangement increases. It will likely be important to many
readers that all subroutines are available for a nominal charge on tape or diskettes
for many computational environments. Fortran 77 is the language of choice for all
subroutines and programs. Reading of programs and subroutines is made easier by
use of mnemonics for subroutine names as well as variable names.
In general, this book is clearly written and well presented with few typographical
errors or awkward constructions. Some readers may dislike use of examples from
both solid and fluid mechanics. Also some examples make use of inefficient analysis
procedures.
The positive aspects of the book far outweigh the negative aspects. It presents
the broad picture of the basis of finite element software very clearly. The detailed
picture of the basis of various finite element concepts is sometimes left incomplete
or fuzzy from the point of view of the novice.
This book will make a fine addition to the library of anyone interested in fi-
nite element computer programming. It is viewed as complementary to the more
764 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
detailed presentations of the finite element concepts found in other texts.
V. J. Meyers
School of Civil Engineering
Purdue University
West Lafayette, Indiana 47907
23[35L65, 35L67, 65N05, 65N30, 76G15, 76H05].—S. P. SPEKREIJSE,
Multigrid Solution of the Steady Euler Equations, CWI Tract 46, Centrum voor
Wiskunde en Informática (Centre for Mathematics and Computer Science), Am-
sterdam, 1988, vi+153 pp., 24 cm. Price Dfl.24.20.
The subject at hand is the numerical solution of the compressible inviscid equa-
tions of fluid dynamics, the Euler equations. Only steady state solutions are desired,
though knowledge of properties of the unsteady Euler equations is required to de-
rive suitable methods. These equations are of great interest in aerodynamics and
turbomachinery applications. One goal of numerical algorithm designers in the field
of computational fluid dynamics is to produce efficient, robust solvers that can be
used as design tools on a routine basis. The designer must be able to vary some
parameter, for example a geometrical quantity, in a specified parameter set, and
study how the solution varies. The millennium is not yet upon us, though it is
perhaps in sight for problems in two space dimensions.
This compact book (153 pages) presents a large amount of material in a remark-
ably clear and concise fashion. From a derivation of the Euler equations and general
material on the Riemann problem for hyperbolic systems in Chapter 1, the author
moves to a long chapter concerned with upwind discretizations of the Euler equa-
tions. The key ingredient is an approximate Riemann solver; the author prefers
Osher's. All the modern apparatus of upwind schemes is clearly presented here:
numerical flux functions, approximate Riemann solvers, total-variation diminish-
ing schemes, monotonicity, and limiters. Noteworthy is the author's new, weaker,
definition of monotonicity for a two-dimensional scheme, which evades the negative
result of Goodman and Leveque that total-variation diminishing schemes in two
space dimensions are at most first-order accurate. The new definition of mono-
tonicity still rules out interior maxima and minima (for scalar equations) but does
not preclude second-order accuracy. The third chapter gives a very brief descrip-
tion of the multigrid algorithm and shows some numerical solutions of a first-order
accurate discretization. Practitioners in the field have found in the past few years
that multigrid algorithms converge well for first-order accurate discretizations of
steady fluid flow equations. The convergence is helped along by a large amount of
numerical viscosity. Unfortunately, this large amount of numerical viscosity leads to
unacceptable inaccuracy in the computed solutions. Multigrid algorithms applied
directly to second-order discretizations (which would give acceptable accuracy) tend
to converge slowly, if they converge at all. This motivates the fourth and last chap-
ter, in which defect correction is used. This allows one to produce a second-order
accurate solution, even though only problems corresponding to first-order accurate
discretizations need to be solved. It seems that this is the first time defect correc-
tion has been used in the solution of any problem in computational fluid dynamics.
Perhaps the use of defect correction here will spark more interest in the technique.
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 765
There are just a few, easily corrected, typographical errors in the displayed
equations of this book. More numerous, though perhaps no more numerous than is
the norm nowadays, are the misspelled English words. The book does not claim to
be a comprehensive account of the numerical solution of the Euler equations, nor
of upwind methods for the Euler equations (central differences and the competitors
to Osher's scheme are not discussed). Statements like "Nowadays, finite volume
schemes are almost universally used for shock capturing codes" could be misleading
to a novice. There are many working codes which are based on finite difference
methods and which produce very good results for problems with shocks. The
expert reader who has had trouble with convergence to steady state may look at
the convergence plots in Chapter 4 of this book and wonder if the algorithm would
produce solutions accurate to machine zero if iterated indefinitely. One who wanted
to play the devil's advocate could look at some of the plots in Chapter 3 and claim
they showed a convergence rate which is not independent of grid size (the method
converging slower on finer grids).
These caveats noted, this book is the best single reference for a person trying
to construct a multigrid code for the steady Euler equations. In this reviewer's
opinion, a novice in the field of computational fluid dynamics, armed only with this
book, could produce a working code. This is a tribute to the book's completeness
and clarity.
Dennis C. Jesperson
NASA Ames Research Center
MS 202A-1Moffett Field, California 94035
24(65-02, 65N30].—D. LEGUILLON & E. SANCHEZ-PALENCIA, Computation
of Singular Solutions in Elliptic Problems and Elasticity, Wiley, Chichester, 1987,
200 pp., 24 cm. Price $42.95.
The monograph is focused on the explicit computation of the singular solutions
near corners and interfaces in plane elasticity theory for multimaterials.
These problems are governed by the elasticity system with piecewise constant
coefficients. Indeed, the constitutive law is different in each component of the
material under consideration. The jumps in the coefficients are across the interfaces
between the different components. These interfaces are usually assumed to be
straight, or polygonal, lines. One can also view these problems as transmission
problems across these interfaces.
The general theory shows that the solutions to these problems behave as
rau(9) + more regular terms
in polar coordinates, where r is the distance to the singular point under considera-
tion, which is either one corner of the boundary, or a corner of an interface, or even
a point where an interface meets the boundary.
766 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
The corresponding stresses behave as rQ_1 and therefore may become infinite
as r goes to 0 when Rea < 1. Although this is in clear contradiction with the
assumption that the strains remain small, which legitimizes the linear theory, it
turns out in practice that it provides a good hint of where damage may occur. The
smaller a is, the likelier is the damage.
The general theory of singular solutions is not provided in this monograph. How-
ever, assuming that the data, volume loads and surface loads, vanish in the vicinity
of the corner or singular point under consideration, the authors give a very straight-
forward and illuminating analysis that shows the form of the singular solution there.
The analysis of the case of one single material shows that a is the solution of
a rather complicated transcendental equation. The corresponding transcendental
equations for multimaterials seem quite out of reach by classical analysis. This
is why it is sensible to rely on numerical computation for producing approximate
values of the corresponding a.
Two approaches to the actual calculation of a are described. They are based on
the fact that the above u is the solution of a second-order boundary value problem
for a system of two ordinary differential equations that depend quadratically on a.
In the first approach, the boundary value problem is discretized directly by the
finite element method using piecewise linear functions. The approximate problem
reduces to finding the zeros of a determinant D(a) that depends analytically on a.
The method is quite cheap, yet accurate.
In the second approach, one performs a preliminary reduction of the order with
respect to a, by writing the boundary value problem as a first-order system. Then
one faces the more common problem of finding eigenvalues and eigenvectors to a
boundary value problem for a system of ordinary differential equations. Again,
this is approximated by piecewise linear elements. One is left with the problem of
finding the eigenelements for a large matrix that is nonsymmetric in most cases.
This method is more expensive and requires heavier equipment, but it also allows
one to calculate u approximately.
The whole monograph is very clearly written. Many illustrative examples are
provided. It will certainly be very helpful to engineers.
Pierre Grisvard
Laboratoire de Mathématiques
Université de Nice
F-06034 Nice Cedex, France
25(49-01, 49H05].—M. J. SEWELL, Maximum and Minimum Principles—A
Unified Approach, with Applications, Cambridge Univ. Press, Cambridge, 1987,
xvi+468 pp., 23 \ cm. Price $79.50 hardcover, $34.50 paperback.
This is a text on operational variational methods. It concentrates on how to
associate extremum principles with differential equations and applied problems.
The book requires a minimal mathematical background and provides a large number
of examples. The book has many exercises, so it could be used as a text in a
mathematical methods course.
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 767
The organization is somewhat unusual. The first chapter treats mostly finite-
dimensional saddle-point problems, with one small section on saddle-point problems
for initial value problems. Chapter 2 is entitled "Duality and Legendre transfor-
mations" and includes some singularity theory and a number of interesting ap-
plications. Again, most of the chapter is finite-dimensional, with some infinite-
dimensional examples interspersed.
The heart of the book is in Chapter 3, where the author shows how to asso-
ciate dual variational problems with a saddle functional. This is done without any
treatment of minimax theorems. Most of the examples involve ordinary differen-
tial equations and include variational inequalities and problems with nonstandard
side conditions. Chapter 4 describes how to derive bounds on linear functionals
of solutions of variational problems and develops some formal methods for finding
variational principles for initial value problems.
The last chapter (Chapter 5) is 90 pages long and is entirely devoted to a theory
of variational principles arising in the mechanics of solids and fluids. Much of
this section reflects the author's own research and is a unique contribution to the
literature.
In the first volume of Courant and Hubert, the calculus of variations was de-
scribed as being calculus on function spaces. Sewell's text provides a calculus, but
not an analysis. It tries to present the theory without using the concepts and tools
of real, or functional, analysis. Such an attitude was necessary in Courant's time. It
is not defensible today. Most of the texts on finite-dimensional optimization theory
referenced here use more analysis than this book does.
Equally surprising for a text in applied mathematics, it does not treat approxi-
mate, or numerical, methods for finding solutions. There is no discussion of second
derivative conditions and no treatment of stability theory. It concentrates entirely
on first derivative conditions and on setting up extremum principles. It does not
study the type of critical point occurring at the solution. For the computational
implementation of variational principles, it is crucial to know the type of a critical
point. Algorithms for computing local minima are based on different considerations
than those for finding saddle points.
This book is not directed at mathematicians. It does not define a Hubert space,
nor does it mention Banach spaces, Lebesgue or Sobolev spaces, or (metric) com-
pleteness. The author ignores many analytical details, and there is no discussion
of existence questions. There is no use of the insights of convex analysis, or of
minimax theory, both of which could have simplified much of the exposition. The
Legendre transformation, described at length in Chapter 2, is a way to compute
the convex conjugate of a given functional in convex analysis. It is the properties
of this conjugate functional, not the Legendre transformation itself, that seem to
be essential in much of duality theory. The author ignores the modern theory of
the calculus of variations and the large volume of literature which has been written
in a more sophisticated mathematical language; especially the contributions of the
French, Italian and Soviet schools. Names that are missing from the bibliography
include those of Brezis, Fichera, Mikhlin and Vainberg.
On the other hand, the examples and applications are of considerable interest.
The book might be an appropriate introduction to these fascinating topics for
768 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
someone, with an interest in applications, who is unwilling to learn elementary
functional analysis. Euler would feel completely at home with this text.
Giles Auchmuty
Department of Mathematics
University of Houston
Houston, Texas 77204-3476
26(65-02, 65Kxx, 90Cxx].—R. FLETCHER, Practical Methods of Optimization,
2nd ed., Wiley, Chichester, 1987, xiv+436 pp., 23 \ cm. Price $53.95.
The development of optimization methods played a particularly vigorous role in
the surge of numerical mathematics following the advent of electronic computing.
Constrained optimization took center stage, first starting in the late 1940's with
the development of linear and nonlinear programming by Dantzig, Chames, Frisch,
Zoutendijk, just to name a few. In the late 1950's Davidon's variable metric method,
its subsequent analysis and independent work by Fletcher and Powell, yielded an
entirely new perspective on the endeavor of unconstrained optimization, which at
the time—hard as it is to comprehend now—was largely considered routine. Both
aspects of optimization have been enriched by progress in numerical linear algebra,
and their methodologies are indeed unthinkable without it.
This flourishing field, as described by one of its foremost pioneers, is the topic
of this meticulously crafted book. It provides a concise, coherent, and no-nonsense
description of the major practical techniques, together with evaluations and rec-
ommendations based on the author's extensive experience. His discussions touch
on many alternatives and extensions to key ideas, placing them also in historical
context. Experience and experiment are stressed throughout, and the text is corre-
spondingly interspersed with illuminating numerical examples. Key theorems are
stated and proved rigorously. A rich variety of exercises at the end of each chapter
deepens the understanding and furnishes important additional detail. The entire
material is supported by a carefully selected, yet complete, bibliography. There
are only very occasional lapses into technical jargon, my favorite being "all pivots
> 0 in Gaussian elimination without pivoting" (page 15). All in all, a highly rec-
ommended reading for those acquainted with the fundamentals of numerical linear
algebra and multivariate calculus.
The division into two parts, on "unconstrained" and "constrained" optimization,
respectively, reflects two different flavors of methodology. Each part had previously
been published as separate volumes [1], [2], with the first part reprinted twice.
The second edition, now combining both volumes, deepens the descriptions of key
subjects in the first part and expands the subject matter of the second. Thus, in the
first part, the Dennis-Moré characterization of superlinear convergence in nonlinear
systems has been included and the important Chapter 2 has been substantially
reorganized. In the second part, a section on "network programming", as well as
a brief discussion of Karmarkar's interior point method, have been added. The
reader will also find many new and stimulating questions in the problem sections.
The second edition offers definitely more than the first edition, while retaining the
focus and the crispness of the previous exposition.
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 769
Chapters 1 through 6 form the first part. The emphasis is on finding "local"
minimizers using local differential information, explicitly in the form of separately
computed gradients, or implicitly by utilizing difference information. The thorny
problem of determining "global" minimizers is only occasionally mentioned. Chap-
ter 1 contains introductory material. The important Chapter 2 surveys the general
structure of the methods and pays well-placed attention to the principles and al-
gorithms for "line search", a feature in many of the subsequently discussed meth-
ods. Chapters 3 and 4 on Newton, quasi-Newton, and conjugate gradient methods
form the core of the first part on unconstrained optimization. The final two chap-
ters round out that material, featuring among others: trust regions, Levenberg-
Marquardt techniques, the Newton-Raphson method and Davidenko continuation
methods for solving systems of nonlinear equations,' the theory of superlinear con-
vergence.
Chapters 7 through 14 form the second part. Here "constrained optimization"
is first and foremost the optimization of functions, generally excluding the large
field of combinatorial optimization. Chapter 7 is again introductory. Chapter 8
features linear programming with emphasis on the simplex method and some of its
variations such as "product form" and ¿[/-factoring of the basis and Dantzig-Wolfe
decomposition. Lagrange multipliers, first- and second-order optimality conditions,
convexity and duality are introduced in Chapter 9 preparatory to the description
of classical nonlinear programming in Chapters 10 through 12: quadratic program-
ming, linearly constrained optimization, zigzagging, penalty and barrier functions,
sequential quadratic programming, feasible directions. Integer programming is only
briefly discussed, stressing branch-and-bound techniques. It shares Chapter 13
with sections on geometric programming and optimal flows in networks. The final
Chapter 14 offers a unique self-contained treatment of nondifferentiable or, rather,
piecewise differentiable optimization.
The book is a classic and invaluable for the practitioner as well as the student
of the field.
Christoph Witzgall
Center for Computing and Applied Mathematics
National Institute of Standards and Technology
Gaithersburg, Maryland 20899
1. R. FLETCHER, Practical Methods of Optimization, Vol. 1: Unconstrained Optimization, Wiley,
Chichester, 1980.
2. R. FLETCHER, Practical Methods of Optimization, Vol. 2: Constrained Optimization, Wiley,
Chichester, 1981.
27(90-01, 90B99].—PANOS Y. PAPALAMBROS & DOUGLASS J. WILDE, Prin-
ciples of Optimal Design—Modeling and Computation, Cambridge Univ. Press,
Cambridge, 1988, xxi+416 pp., 26 cm. Price $49.50.
For anyone interested in modeling, model building, and in particular, optimiza-
tion models and the interaction between optimization and the modeling process,
this book is a must. It combines classical optimization theory with new ideas of
770 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
monotonicity and model boundedness that provide valuable information for deter-
mining the most efficient and correct formulation of a model. It is an easy-to-read
book with clear, modern notation, well-drawn graphs, and an easy-to-follow orga-
nization. It contains just enough theory and proofs to be complete, without being
overburdened, and follows each concept with examples, applications, and realis-
tically designed engineering design problems. Although aimed at the engineering
design student, it is a valuable addition to the libraries of operations researchers,
economists, numerical analysts, and computer scientists. The engineering flavor
might at first be discouraging, but the discomfort quickly vanishes as the down-to-
earth treatment of the topics comes through.
There are eight chapters in the book. Each has a concise introduction to set the
stage for the concepts to be presented, and a summary to tie all the ideas together
and reinforce what was introduced. The chapters are organized so as to create in
the reader an appreciation for models, from their definition to their use.
Chapter 1 is an excellent introduction to mathematical modeling and the differ-
ent types of models. It defines the design optimization problem and most of the
other ideas and issues that are addressed throughout the book, including feasibility
and boundedness, design space topography, data modeling, solution, and computa-
tion. The concept of a "good" model is introduced (one that represents reality in
a simple but meaningful manner), and the limitations of modeling are also touched
on.
The second chapter, on model boundedness, introduces the concepts of verifica-
tion and simplification, which all too often are overlooked in modeling literature
and practice. The authors stress the importance of applying the techniques of this
chapter before performing any detailed computation, and note that in so doing, one
obtains reductions in model size, tighter formulations, more robust models, and
even formulation error detection. In this, the authors could not be more correct.
As computers and computing become cheaper and more powerful, more researchers
are relying on modeling and computational experiments to test new ideas. Problems
which could not be solved before are now being addressed with success. The ideas
introduced in this chapter are critical to this process and are the same ones used so
successfully, for example, to solve large-scale integer programming problems. (See,
e.g., [1], [2].)
Chapter 3, on interior optima, provides the mathematical foundation for local
iterative methods. Its concise treatment is especially useful for those who want
not only to understand the theory but also to use the methods to solve problems.
Optimality conditions are presented here, along with convexity, gradient methods,
Newton-type methods, and stabilization using modified Cholesky factorization. Un-
fortunately, there is no discussion about interior point methods for solving linear
programming problems.
Boundary optima are the subject of Chapter 4. The discussion proceeds from
feasible directions to sensitivity analysis. Also included are discussions of reduced
gradients, Lagrange multipliers, the generalized reduced gradient method, gradient
projection, and Karush-Kuhn-Tucker conditions. After the general case is treated,
linear programming is developed at the end of the chapter, an approach I found
appealing.
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 771
In Chapter 5, on model reduction, the ideas from Chapters 2, 3, and 4 are
combined into a well-developed approach to the idea of producing as tight a for-
mulation as is possible. Two new concepts are introduced to help in this search
for a minimal formulation, a rigorous Maximal Activity Principle, and a heuris-
tic Coincidence Rule. I found this theoretical approach to model reduction very
appealing.
Global bound construction, the topic of Chapter 6, is the logical next step in
the progression toward developing useful models. Constructing tight bounds, as
the authors note, not only saves effort, but avoids accepting suboptimal solutions.
Many techniques for constructing bounds are introduced here, from simple lower
bounds to geometric inequalities, to unconstrained geometric programming. Com-
bining these techniques into a process for model reduction using branch and bound
is also discussed.
Chapter 7, on local computation, contains descriptions of various numerical al-
gorithms for solving nonlinear programming problems. It is introduced with a brief
section on choosing a method from among the large number of generally accepted
techniques that are available. It contains sections on convergence, termination cri-
teria, single variable minimization, Quasi-Newton methods, differencing, scaling,
active set strategies, and Penalty and Barrier methods.
The real action is in Chapter 8, Principles and Practice. It begins with a review
of the modeling techniques and approaches presented in the previous chapters. The
goal is "to point out again the intimacy between modeling and computation that
was explored first in Chapter 1." An optimization checklist is also provided for the
novice modeler to use as a prompt while gaining more experience.
Richard H. F. JacksonCenter for Manufacturing Engineering
National Institute of Standards and Technology
Gaithersburg, Maryland 20899
1. K. L. HOFFMAN & M. W. PADBERG, Techniques for Improving the Linear Programming
Representation of Zero-One Programming Problems, George Mason Univ. Tech. Rep., 1988.
2. H. CROWDER, E. L. JOHNSON & M. PADBERG, "Solving large-scale zero-one linear pro-
gramming problems," Oper. Res., v. 31, 1983, pp. 803-834.
28(65-01, 65Fxx].—Dario Bini, Milvio Capovani & Ornella Menchi,
Metodi Numerici per l'Algebra Lineare, Zanichelli, Bologna, 1988, x+514 pp., 24
cm. Price L. 48 000 paperback.
An excellent treatment of numerical methods in linear algebra, this text is des-
tined to become the standard work on the subject in the Italian language. It
contains seven chapters, of which the first three are introductory, providing the
necessary tools of matrix algebra, eigenvalue theory and estimation, and norms.
Chapter 4 discusses the major direct methods, Chapter 5 the more important iter-
ative methods, for solving linear algebraic systems. Methods for computing eigen-
values and eigenvectors of symmetric and nonsymmetric matrices are the subject of
Chapter 6. The final chapter deals with least squares problems and related matters.
772 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
The exposition, throughout, is crisp and to the point. There is a healthy balance
between theoretical analysis, the study of error propagation and complexity, and
practical experimentation. Each chapter is provided with a superb collection of
exercises—many supplied with solution sketches—and with historical notes and
bibliographies. For future editions, however, the authors may wish to pay more
attention to the correct spelling of names.
W. G.
29[68T99, 65D07, 65D10].—ANDREW BLAKE & ANDREW ZlSSERMAN, Vi-
sual Reconstruction, The MIT Press Series in Artificial Intelligence, The MIT
Press, Cambridge, Mass., 1987, ix+225 pp., 23 \ cm. Price $25.00.
This book appears in a series on artificial intelligence, and seems to be aimed
primarily at computer scientists. However, mathematicians interested in computer
vision will certainly want to look at it, and it may be of some peripheral interest to
numerical analysts and approximation theorists interested in optimization and/or
splines.
The subject is visual reconstruction, which is defined by the authors to be the
process of reducing visual data to stable descriptions. The visual data may be
thought of as coming from photoreceptors, spatio-temporal filters, or from depth
maps obtained by stereopsis or optic rangefinders. Stability in this setting refers to
the desire that the representation should be invariant to certain distortions such as
sampling grain, optical blurring, optical distortion and sensor noise, rotation and
translation, perspective distortions, and variation in photometric conditions.
The bulk of the book is devoted to the use of certain variational methods (called
here weak strings, weak rods, and weak plates) for detecting edges (discontinuities
in value or slope) of functions and surfaces. The methods are a form of penalized
least squares, where the penalty includes a measure of smoothness of the function
(typically an integral of a derivative) which is reminiscent of spline theory. These
problems are analyzed using variational methods, and solved by discretizing them
and applying an appropriate optimization method. The optimization method dis-
cussed here is referred to as the graduated non-convexity algorithm, and is designed
to work on the nonconvex problems arising here. Convergence properties and the
optimality of the algorithm are discussed.
The book includes about 150 references, mostly in the computer science liter-
ature. The authors seem to assume that the reader is familiar with much of this
literature; results are often referred to without explanation. Readers not familiar
with such things as "pontilliste depth map", "hyperacuity", "cyclopean space",
or "data-fusion machine" may find the going hard. The authors have elected to
"avoid undue mathematical detail", and despite several appendices, I expect that
most mathematicians will not be fully satisfied.
L.L. S.
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 773
30[01A45]. A. W. F. EDWARDS, Pascal's Arithmetical Triangle, Oxford Univ.
Press, New York, 1987, xii+174 pp., 24 cm. Price $37.50.
As an impressive culmination of meticulous research into original sources, this
definitive study constitutes the first full-length history of the Arithmetical Triangle,
reputedly the most celebrated of all number patterns. Its origins are herein traced
to Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra.
The elements of this triangle evolved historically in three equivalent forms;
namely, figúrate numbers, combinatorial numbers, and binomial coefficients. The
different aspects of these numbers were first combined by Blaise Pascal in his Traité
du triangle arithmétique, written in 1654 and printed posthumously in 1665. He was
the first to consider the properties of this array as pure mathematics, deduced from
the fundamental addition relation independently of any binomial or combinatorial
application.
Dr. Edwards discusses in detail this creative treatise and shows, in particular,
how it influenced Wallis, Newton, and Leibniz in the early development of analysis.
Special attention is given to Wallis's ingenious derivation of his well-known infinite
product by means of interpolation in an analogous triangular array of reciprocals
of figúrate numbers.
The modern theory of probability originated in the extensive correspondence
between Pascal and Fermât concerning the previously unsolved problems called,
respectively, the Problem of Points and the Gambler's Ruin. The solutions obtained
by Pascal and Fermât have been described in detail in two previously published
papers by the author, which are herein reprinted as appendices. Dr. Edwards
establishes that the basic solution of the first problem is properly attributable to
Pascal instead of Fermât, contrary to tradition.
The origins of the binomial and multinomial distributions are also traced to
Pascal's treatise, as revealed in the correspondence of Leibniz, John Bernoulli,
De Moivre, and Montmort.
A concluding chapter is devoted to a discussion of James Bernoulli's famous
work, Ars conjectandi, which is chiefly noted for containing the first limit theorem
in probability. Apparently, James Bernoulli independently derived the binomial
distribution, unaware of Pascal's treatise, to which his attention was first drawn
just before his death in 1705 by Leibniz.
Each of the ten chapters of this carefully written book is supplemented by ex-
planatory notes and bibliographic references. In addition, there is a comprehensive
list of nearly two hundred sources and an index of the names of more than one
hundred persons appearing in this historical account.
This scholarly book should be of special interest to students and teachers of the
history of mathematics and statistics.
J. W. W.
774 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS
31[65F10, 65N20].—A. HADJIDIMOS (Editor), Iterative Methods for the Solu-
tion of Linear Systems, North-Holland, Amsterdam, 1988, 291 pp., 27 cm. Price
$78.00/Dfl. 160.00.
This is a reissue in book form of Journal of Computational and Applied Math-
ematics, v. 24, 1988, nos. 1 & 2. It provides a useful cross section of current
work in the area of the title, including topics such as convergence theory, parame-
ter selection, preconditioning, rectangular and infinite systems, implementation on
machines with parallel architectures, and software packages.
W. G.
32(65-06, 68-06].—JAMES McKENNA & ROGER TEMAM (Editors),
ICIAM '87: Proceedings of the First International Conference on Industrial
and Applied Mathematics, SIAM, Philadelphia, 1988, xx+376 pp., 26 cm. Price
$56.50.
The conference in the title of these proceedings, co-organized by four sister soci-
eties of applied mathematics, GAMM, IMA, SIAM and SMAI from Germany, Eng-
land, the United States and France, was held in Paris on June 29—July 3, 1987.
Part I contains the welcoming and opening addresses. Sixteen invited papers make
up Part II. The authors and their titles are: Wolfgang Alt, Modelling of motility in
biological systems; Karl Johan Aström, Stochastic control theory; Michael Atiyah,
Topology and differential equations; Robert Azencott, Image analysis and Markov
fields; Philippe G. Ciarlet, Modeling and numerical analysis of junctions between
elastic structures; D. G. Crighton, Aeronautical acoustics: mathematics applied to
a major industrial problem; Carl de Boor, What is a multivariate spline?; Yves
Genin, On a duality relation in the theory of orthogonal polynomials and its ap-
plication in signal processing; W. Hackbusch, A new approach to robust multi-grid
solvers; John Hopcroft, Model driven simulation systems; Peter D. Lax, Mathemat-
ics and computing; L. Lovász, Geometry of numbers: an algorithmic view; Andrew
Majda, Vortex dynamics: numerical analysis, scientific computing, and mathemat-
ical theory; F. Natterer, Mathematics and tomography; P. Perrier, Numerical flow
simulation in aerospace industry; M. J. D. Powell, A review of algorithms for nonlin-
ear equations and unconstrained optimization. A similar variety of topics is treated
in 69 minisymposia whose abstracts appear in Part III. A list of contributed papers
in Part IV, and an author index, conclude the volume. Full-length versions of 29
contributions by Dutch authors have previously been published in [1].
W. G.
1. A. H.P. VAN DER BURGH & R. M.M. Mattheu (Editors), Proceedings of the First In-
ternational Conference on Industrial and Applied Mathematics (ICIAM 8T): Contributions from the
Netherlands, CWI Tract, Vol. 36, Centre for Mathematics and Computer Science, Amsterdam,
1987. [Review 21, Math. Comp., v. 50, 1988, p. 649.]
REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 775
33(65-06]. ROLAND GLOWINSKI, G ENE H. GOLUB, GÉRARD A. MEURANT
&¿ JACQUES PeriAUX, First International Symposium on Domain Decompo-
sition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988,
x+431 pp., 26 cm. Price $48.50.
This is the proceedings of a symposium on the subject of the title of this volume.
It consists of 22 papers by invited speakers. The papers presented were not subject
to the refereeing process. The topics of the articles include theoretical foundations,
applications to physical problems, related techniques such as block relaxation and
computer implementation of decomposition algorithms.
J.H.B.
34(33-06, 33A65, 41-06, 42C05].—M. ALFARO, J. S. DEHESA, F. J.
MARCELLAN, J. L. RUBIO DE FRANCIA & J. VINUESA (Editors), Orthog-
onal Polynomials and Their Applications, Lecture Notes in Math., vol. 1329,
Springer-Verlag, Berlin, 1988, xv+334 pp., 24 cm. Price $28.60.
These are the proceedings of the Second International Symposium on Orthogo-
nal Polynomials and their Applications held in Segovia, Spain, September 22-27,
1986. (For the first symposium, see [1].) The volume contains nine invited lectures,
covering analytic and group-theoretic aspects of orthogonal polynomials and appli-
cations to approximation theory. In addition, there are 13 contributed papers and
a collection of open problems.
W.G.
1. C. Brezinski, A. Draux, A. P. Magnus, P. Maroni & A. Ronveaux (eds.), Polynômes
Orthogonaux et Applications, Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985. (Review
23, Math. Comp., v. 49, 1987, pp. 305-306; Corrigendum, ibid., v. 50, 1988, p. 359.)
35(65-06, 68-06].—GARRY RODRIGUE (Editor), Parallel Processing for Scien-
tific Computing, SIAM, Philadelphia, 1989, 428 pp., 25 \ cm. Price $45.00.
These are the proceedings of the Third SIAM Conference on Parallel Process-
ing for Scientific Computing held in Los Angeles, California, December 1-4, 1987.
The papers (and in some cases abstracts only) of 14 invited lectures and 76 con-
tributed talks are arranged in seven parts: Matrix Computations, Numerical Meth-
ods, Differential Equations, Scientific Applications, Languages, Software Systems,
and Architectures.
W. G.